# Heat Transfer Through a Plane Wall

1. Sep 16, 2011

### bob1182006

1. The problem statement, all variables and given/known data
Consider a large plane wall of thickness L = 0.4m, thermal conductivity k = 1.8W/(m*K), and surface area A = 30m2. The left side of the wall is maintained at a constant temperature of T1 = 90 C while the right side looses heat by convection to the surrounding air at Ts = 25 C with a heat transfer coefficient of h = 24 W(m2*K). Assuming constant thermal conductivity and no heat generation in the wall evaluate the rate of heat transfer through the wall.

2. Relevant equations
$$\dot{Q}_{wall} = -kA\frac{dT(0)}{dx}$$
$$\frac{d^2T}{dx^2} = 0$$
$$T(0) = 90$$
$$-k\frac{dT(L)}{dx} = h[T(L) - Ts]$$

3. The attempt at a solution

Solving the differential equation and applying B.C.:
$$T(x) = xC_1 + C_2$$
$$T(0) = C_2 = 90$$
$$-kC_1 = hLC_1+hC_2-25h]$$
$$C_1 = -\frac{h(C_2-25)}{k+hL}$$

Plugging numbers in:
$$C_2 = 90$$
$$C_1 = -136.8$$
$$T(x) = 90-136.8x$$
$$\dot{Q}_{wall} = -1.8*30*(-136.8) = 7,387 W$$

Did I make a mistake or is the book's answer wrong?

2. Sep 17, 2011

### rude man

I used a somewhat different approach (underlying equations are of course the same) and got the same answer you did.

I computed the thermal resitance of the wall as L/kA = 7.41e-3 K/W and the equivalent thermal resistance of the convection effect as 1/hA = 1.39e-3 K/W, giving total thermal resistance of 8.80e-3 K/W so dQ/dt = (90 - 25)/8.80e-3 = 7387W.

Just curious - what is your textbook?

3. Sep 17, 2011

### bob1182006

Awesome thanks, I didn't want to waste any more time redoing this problem looking for a mistake, and my class is just starting the thermal resistance chapter so if I see more problems that have wrong answers I'll do them that way to check.

We're using Heat and Mass Transfer Fundamentals & Applications 4th Ed by Cengel and Ghajar.

4. Oct 18, 2011

### spree

arent we suppose to take temp. as kelvin here?

5. Oct 18, 2011

### rude man

It's OK to interchange K and C as long as you're dealing with temperature differences.

6. Oct 18, 2011

### spree

oh ure right. new to the subject, just trying to learn sorry :) btw I personally know professor cengel and i will inform him about this. I will let you know about it. if you suspect any other mistake in the book feel free to ask please.